Distribution of inactive chains

Thus f, (m) is the (unnormalized) length distribution of inactive chains formed by disproportionation, particularly in systems where disproportionation represents an exclusive or predominating termination mechanism. f2(m) corresponds to the (unnormalized) length distribution of macroradicals. 

Combination of Equations 6.25 and 6.26 yields that is the generation rate of the fcth order moment of the distribution of inactive chains (see Section 2.3 for details about the calculation of the moments) ... 

Many of the proteins of membranes are enzymes. For example, the entire electron transport system of mitochondria (Chapter 18) is embedded in membranes and a number of highly lipid-soluble enzymes have been isolated. Examples are phosphatidylseiine decarboxylase, which converts phosphatidylserine to phosphatidylethanolamine in biosynthesis of the latter, and isoprenoid alcohol phosphokinase, which participates in bacterial cell wall synthesis (Chapter 20). A number of ectoenzymes are present predominantly on the outsides of cell membranes.329 Enzymes such as phospholipases (Chapter 12), which are present on membrane surfaces, often are relatively inactive when removed from the lipid environment but are active in the presence of phospholipid bilay-ers.330 33 The distribution of lipid chain lengths as well as the cholesterol content of the membrane can affect enzymatic activities.332... 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

By fractionation of products formed at various conversions and on multiple addition of monomer other facts could be established. Thus the bimodal distribution is formed early in the reaction and persists throughout. Growing polymer chains of all chain lengths exist with the possible exception of the petroleum ether-soluble material, which is essentially inactive at least by the end of polymerization. The longer chains seem to be adding monomer faster. 

The potassium silanolate exists largely as the associated species 2, which is considered unreactive. The molecular weight distribution inevitably shows a low-molecular-weight tail when these catalysts are used and is attributed to the fraction of the chain ends in the inactive form 2 (37). 

Zak and Frank [41] are pioneers who used discrete Markov Chain in finite state to describe a stochastic nature of supply and demand under centralized distribution of resources. The outcome shows that the size of system loss has close relations with functions of the system, the initial inventory level and allocation of resourees. To avoid material shortage and resources inactiveness, they designed mathematical models in different situations and algorithms applying brand and bound method to optimize the initial inventory and resource allocation. The goal of those measures was to minimize total system costs of losses. 

Size distributions of P(L-HMPMAm) are bimodal below Tcp (6°C) P(DL-HMPMAm) is optically inactive possesses better solubility in water than P(L-HMPMAm) it exhibits some degree of turbidity at 34°C, but the transmittance does not decrease to 0% no hysteresis P(DL-HMPMAm) forms a clear coacervate above 34°C forms no solid precipitate steiic hindrance between the side chains (racemic monomers) results in relatively expanded stiuctures of polymeric chains above 34°C... 

Let Pi (l)u be the volume fraction of inactive material and let P2(N)v denote the dimensionless concentration of activated Ai-mers (u is the monomer volume). The grand potential of the inactive monomers isgivenby= pi(l)[ln pi(1)l> - 1 - lnZi(l) - /Sp,], while that of the active ones, Q.2, is given by Eq. (1), where as before we replace p N) by P2 N). The equilibrium distribution of material over the active and inactive states is determined by the set of equations 5f2/5pi(l) = 0 and SQ/Sp2 N) = where = f2i - - Here, i/ acts as a Lagrange multiplier (or chemical potential) that keeps the number of active chains constant. 

The previous analysis allows determination of dp using the kinetic parameters in a steady-state polymerization however, a complete characterization of the molecular weight distribution requires the first three moments of the chain length distribution (to provide M, and PDI). A statistical approach to analyze the inactive polymer chains can be used to calculate these quantities. 

For entries 3-5 the increase in molecular weight observed can be assigned to the increase in the rate of insertion and the rate of termination remains practically the same. An increase of the rate of polymerisation with the steric bulk of the ligand is usually ascribed to the destabilisation of the alkene adduct while the energy of the transition state remains the same. As a chain transfer reaction presumably P-hydride elimination takes place or traces of water might be chain transfer agents. Chain transfer does occur, because a Schulz-Flory molecular weight distribution is found (PDI 2, see Table 12.2). Shorter chains are obtained with a polar ortho substituent (OMe, entry 2) and in methanol as the solvent, albeit that most palladium is inactive in the latter case. 

The properties of the isolated peptides were quite similar in nature, whereby each peptide consisted of 12 amino acids in length and possessed a munber of residues with functional side groups that could stabilize nanoclusters. In many instances, these side chains were the hydroxyl-terminated side chains of serine, threonine, and tyrosine. In two of the peptides (AG3 and AG4), the location of the hydroxylated amino acids was conserved within two of the peptides. Similarly, one proline amino acid was conserved throughout all three of the sequences. Upon incubating each peptide in a solution of silver nitrate with no exogenous reductant, a clearly observable plasmon resonance peak arose at 440 nm for AG3 and AG4, but not with AG5. The peak was quite broad, indicative of a disperse size and shape distribution. The main difference between the active peptides and inactive AG5 was an overall basic isoelectric point for AGS The assays were performed at neutral conditions which would modulate the side-chain dynamics under acidic or basic conditions. 

More recently it has been shown that in the polymerization with TT-crotylnickel iodide the order in monomer falls from a value close to unity at [M] below 0.5 mole 1" to below 0.5 at [M] > 4 mole 1 . These observations have been interpreted in terms of scheme (c) on p. 162, namely coordination of two monomer molecules with the catalyst and with most of the catalyst existing in the complex (inactive) state. The molecular weights of the polymers are double those calculated from the kinetic scheme put forward [61] and this is attributed to coupling of live polymer chains on termination [251]. Molecular weight distributions are binodal consistent with slow propagation and transfer. 

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